Unified approach to the representations of groups SU(2) and SU(1, 1)

1970 ◽  
Vol 48 (10) ◽  
pp. 1272-1282
Author(s):  
Klang-Chuen Young
Keyword(s):  
Generalized Functions ◽  
Irreducible Representations ◽  
Matrix Elements ◽  
Unified Approach ◽  
Canonical Bases ◽  
Representations Of Groups ◽  
Special Cases ◽  
The Matrix ◽  
Finite Transformation ◽  
Explicit Expressions

A unified approach to the representations of groups SU(2) and SU(1, 1) is made. The method is based on the observation that SU(2) and SU(1, 1) can be considered as special cases of a group G(a). The representation of G(a) is realized in the space of homogeneous generalized functions. The canonical bases of the unitary irreducible representations are constructed explicitly. The matrix elements for the finite transformation are found. Explicit expressions for the Wigner coefficients are also obtained.

scholarly journals Corrections to Wigner–Eckart Relations by Spontaneous Symmetry Breaking

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1120
Author(s):  
Carlo Heissenberg ◽  
Franco Strocchi
Keyword(s):  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking ◽  
Irreducible Representations ◽  
Matrix Elements ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Goldstone Bosons ◽  
The Matrix ◽  
Symmetry Breaking Term ◽  
Breaking Term

The matrix elements of operators transforming as irreducible representations of an unbroken symmetry group G are governed by the well-known Wigner–Eckart relations. In the case of infinite-dimensional systems, with G spontaneously broken, we prove that the corrections to such relations are provided by symmetry breaking Ward identities, and simply reduce to a tadpole term involving Goldstone bosons. The analysis extends to the case in which an explicit symmetry breaking term is present in the Hamiltonian, with the tadpole term now involving pseudo Goldstone bosons. An explicit example is discussed, illustrating the two cases.

scholarly journals Matrix Elements of the Boltzmann Collision Operator in a Basis Determined by an Anisotropic Maxwellian Weight Function including Drift

1980 ◽  
Vol 33 (2) ◽  
pp. 449 ◽  
Author(s):  
Kailash Kumar
Keyword(s):  
Weight Function ◽  
Cross Sections ◽  
Collision Operator ◽  
Weight Functions ◽  
Anisotropic Pressure ◽  
Matrix Elements ◽  
Methods Of Calculation ◽  
Special Cases ◽  
The Matrix ◽  
Boltzmann Collision Operator

The matrix elements of the linear Boltzmann collision operator are calculated in a Burnett-function basis determined by a weight function which itself describes a velocity distribution with a net drift and an anisotropic pressure (or temperature) tensor. Three different methods of calculation are described, leading to three different types of formulae. Two of these involve infinite summations, while the third involves only finite sums, but at the cost of greater complications in the summands and the integrals over cross sections. Both elastic and inelastic collisions are treated. Special cases arising from particular choices of the parameters in the weight functions are pointed out. The structure of the formulae is illustrated by means of diagrams. The work is a contribution towards establishing efficient methods of calculation based upon a better understanding of the matrix elements in such bases.

scholarly journals Transfer scattering matrix of non-uniform surface acoustic wave transducers

1987 ◽  
Vol 10 (3) ◽  
pp. 563-581
Author(s):  
N. C. Debnath ◽  
T. Roy
Keyword(s):  
Acoustic Wave ◽  
Equivalent Circuit ◽  
Surface Acoustic Wave ◽  
Scattering Matrix ◽  
Circuit Model ◽  
Complex Frequency ◽  
Matrix Elements ◽  
Special Cases ◽  
The Matrix ◽  
Scattering Matrix Elements

This paper is concerned with a general mathematical theory for finding the admittance matrix of a three-port non-uniform surface acoustic wave (SAW) network characterized bynunequal hybrid sections. The SAW interdigital transducer and its various circuit model representations are presented in some detail. The Transfer scattering matrix of a transducer consisting ofNnon-uniform sections modeled through the hybrid equivalent circuit is discussed. General expression of the scattering matrix elements for aN-section SAW network is included. Based upon hybrid equivalent circuit model of one electrode section, explicit formulas for the scattering and transfer scattering matrices of a SAW transducer are obtained. Expressions of the transfer scattering matrix elements for theN-section crossed-field and in-line model of SAW transducers are also derived as special cases. The matrix elements are computed in terms of complex frequency and thus allow for transient response determinations. It is shown that the general forms presented here for the matrix elements are suitable for the computer aided design of SAW transducers.

The matrix elements of tensor operators for configurations of three equivalent electrons

1959 ◽  
Vol 250 (1263) ◽  
pp. 562-574 ◽  
Keyword(s):  
Irreducible Representations ◽  
Matrix Elements ◽  
Irreducible Tensor ◽  
Quantum Numbers ◽  
Fractional Parentage ◽  
The Matrix ◽  
Tensor Operators

The formulae of Redmond are used to construct expressions for the fractional parentage coefficients relating the configurations l 3 and l 2 . The explicit occurrence of godparent states is avoided for the quartet states of f 3 and also for a sequence of doublet states. The latter are defined by the set of quantum numbers f 3 WUSLJJ 2 , where W and U are irreducible representations of the groups R 7 and G 2 . Matrix elements of the type ( f 3 WUSL || U k || f 3 W'U'SL' ), where U k is the sum of the three irreducible tensor operators u k corresponding to the three f electrons, are tabulated for k = 2, 4 and 6 and for all values of W, U, S and L .

On the direct calculations of the representations of the three-dimensional pure rotation group

1964 ◽  
Vol 60 (1) ◽  
pp. 61-66 ◽  
Author(s):  
Yehiel Lehrer-Ilamed
Keyword(s):  
Finite Elements ◽  
Direct Method ◽  
Three Dimensional ◽  
Rotation Group ◽  
Irreducible Representations ◽  
Matrix Elements ◽  
Pure Rotation ◽  
The Matrix ◽  
Explicit Formulae ◽  
Direct Calculations

AbstractExplicit formulae are given for calculating the matrix elements of the irreducible representations of the three-dimensional pure rotation group by the direct method. In addition explicit formulae are derived to calculate the representations of the finite elements of any group when the eigenvalues of the matrix representing the corresponding infinitesimal elements are given.

scholarly journals GENERALIZATION OF THE GELL–MANN DECONTRACTION FORMULA FOR sl(n,ℝ) AND su(n) ALGEBRAS

2011 ◽  
Vol 08 (02) ◽  
pp. 395-410 ◽  
Author(s):  
IGOR SALOM ◽  
DJORDJE ŠIJAČKI
Keyword(s):  
Closed Form ◽  
Lie Algebra ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Algebraic Expression ◽  
Matrix Elements ◽  
Group Manifold ◽  
The Matrix

The so-called Gell–Mann or decontraction formula is proposed as an algebraic expression inverse to the Inönü–Wigner Lie algebra contraction. It is tailored to express the Lie algebra elements in terms of the corresponding contracted ones. In the case of sl (n,ℝ) and su (n) algebras, contracted w.r.t. so (n) subalgebras, this formula is generally not valid, and applies only in the cases of some algebra representations. A generalization of the Gell–Mann formula for sl (n,ℝ) and su (n) algebras, that is valid for all tensorial, spinorial, (non)unitary representations, is obtained in a group manifold framework of the SO(n) and/or Spin (n) group. The generalized formula is simple, concise and of ample application potentiality. The matrix elements of the [Formula: see text], i.e. SU(n)/SO(n), generators are determined, by making use of the generalized formula, in a closed form for all irreducible representations.

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

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